We are asking for an advice on how to compute a standard deviation of
vector direction. The methods of circular statistics are not appropriate
here because they neglect the vector magnitude. Note that the directions of
the individual vectors do not prescribe precisely the direction of the
resultant (average) vector. While the direction of the resultant (average)
vector can be easily determined by vector averaging, the standard deviation
is different. Here is an example. Suppose we have three force vectors
concurrent at a point; their magnitudes and directions are (1N, 0 deg), (5
N, 45 deg) and (12 N, 90 deg). What is the standard deviation of the force
direction? Note that the direction of the resultant (average) vector is not
45 deg.

As an aside, is it valid to assume that the average vector is in the same
direction as the resultant vector?